Disable showing of run window in IntelliJ

These are called Tool Windows. Right-click the tool window, it pops up a menu, the last option of which is 'Hide'.

Alternatively,

You can install the plugin ToolWindow Manager to show/hide the Tool Window of your choice.

There is currently no solution to do this permanently. However, there is a workaround to prevent it from popping up. Try this:

After the first time the window pops-up simply resize it all the way to the bottom (i.e. grab the TOP and drag down until the window disappears). This will prevent it from popping up again until you manually click one of the buttons to open any window in the same area (i.e. Run, TODO, Android, Terminal, Messages, (and Find Results) - or whatever you have in that same area if you've customized it).

To keep it from popping up, never click the hide button on any of these windows but rather use the same "drag from top to bottom" to hide them (the effect is the same).

This is annoying, to say the least, but once you get used to it it works fine and is much less annoying than dealing with that blasted Run popup window :)

NOTE
I should mention that this is completely safe to do as the windows are only hidden when you do this, and clicking the window's button at the bottom will show it in its original size - so nothing goes missing or breaks using this method :-)

I finally could solve this one. At least it worked for me. At the bottom of "Run/Debug configurations" window and in "Before launch" part, uncheck "Activate tool window" check box.